Fully nonlinear simple internal waves over subcritical slopes in continuously stratified fluids: Theoretical development

The dynamics of fully nonlinear internal waves in continuously stratified fluids is investigated using a new simple-wave (or Riemann-wave) theory with 2nd-order topographic effects. Unlike previous fully nonlinear internal-wave theories, the proposed theory is applicable to large water-depth change over a long distance, provided that the local bottom slopes are mild. An essential step in the theoretical development is the separation of “adiabatic” and “diabatic” topographic effects. First, adiabatic topographic effects are incorporated into the previous simple-wave theory, in such a way that one-directional, fully nonlinear, nondispersive waves become conservative over topography. Since the Riemann variable (or invariant) remains constant along the characteristic, this extends the applicability of the solution method based on characteristic curves from flat bottom to mild slopes. Then, 2nd-order diabatic topographic effects are added to the adiabatic simple-wave solution by a perturbation approach. The application of the proposed theory to Wunsch’s subcritical wedge problem suggests that the proposed theory is applicable to subcritical slopes, although the error unavoidably increases as the slope angle approaches the propagation angle of internal wave rays (i.e., the critical angle). An important finding is the relatively rapid growths of wave-induced isopycnal setup and mean flow under the combined effects of strong nonlinearity and topography, in contrast to temporally stationary setup and mean flow under weak topographic effects. This implies, for example, that large-amplitude internal waves on continental shelves have an inherent tendency to modify the “background” stratification and currents without mixing and that it could occur within a spring-neap tidal cycle.